A note on the k-tuple domination number of graphs

Martínez, Abel Cabrera

doi:10.26493/1855-3974.2600.dcc

Cited by 4 publications

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“…Recently, Hansberg and Volkmann [9] put into context all relevant relationships concerning k-tuple domination (with emphasis in the case k = 2) that have been found up to 2020. Subsequently, Cabrera-Martínez [2,3] obtained new results in this direction. In particular, the following theorem solved an open problem posed in [9].…”

Section: Relationships With Other Domination Parametersmentioning

confidence: 94%

“…Observe that the 1-tuple dominating set of G is the same as the dominating set of G. The k-tuple domination number of G, denoted by γ ×k (G), is the minimum cardinality among all k-tuple dominating sets of G. For a comprehensive survey on k-domination and k-tuple domination in graphs, we suggest the chapter [9] due to Hansberg and Volkmann. In addition, some recent results on these parameters can be found in [1,2,5,13,16].…”

Section: Introductionmentioning

confidence: 99%

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Some new results on the k-tuple domination number of graphs

Martínez

2022

RAIRO-Oper. Res.

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Let $k\geq 1$ be an integer and $G$ be a graph of minimum degree $\delta(G)\geq k-1$. A set $D\subseteq V(G)$ is said to be a $k$-tuple dominating set of $G$ if $|N[v]\cap D|\geq k$ for every vertex $v\in V(G)$, where $N[v]$ represents the closed neighbourhood of vertex~$v$. The minimum cardinality among all $k$-tuple dominating sets is the $k$-tuple domination number of $G$. In this paper, we continue with the study of this classical domination parameter in graphs. In particular, we provide some relationships that exist between the $k$-tuple domination number and other classical parameters, like the multiple domination parameters, the independence number, the diameter, the order and the maximum degree. Also, we show some classes of graphs for which these relationships are achieved.

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Section: Relationships With Other Domination Parametersmentioning

confidence: 94%

Section: Introductionmentioning

confidence: 99%

Some new results on the k-tuple domination number of graphs

Martínez

2022

RAIRO-Oper. Res.

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“…F. Harary and T.W. Haynes in [10,11] introduced double domination, which generalises domination in graphs, and more generally, the concept of k-tuple domination, which has been studied in [12,13] too. Let k be a positive integer.…”

Section: Introductionmentioning

confidence: 99%

On Proper 2-Dominating Sets in Graphs

Bednarz,

Pirga

2024

Symmetry

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This paper introduces the concept of proper 2-dominating sets in graphs, providing a comprehensive characterisation of graphs that possess such sets. We give the necessary and sufficient conditions for a graph to have a proper 2-dominating set. Graphs with proper dominating sets can have a symmetric structure. Moreover, we estimate the bounds of the proper 2-domination number in the graphs with respect to the 2-domination and 3-domination numbers. We show that the cardinality of γ2¯-set is greater by one at most than the cardinality of γ2-set.

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“…The maximum number of classes of a domatic partition (idomatic partition) of G is called the domatic number (idomatic number) of G, denoted by d(G) (id(G)) The concepts of domination and domatic partition and their variations have been studied widely in the literature. See, for example, [1,2,3,4,5,6,12].…”

Section: Introductionmentioning

confidence: 99%

Independent coalition in graphs: existence and characterization

Samadzadeh,

Mojdeh

2024

Ars Math. Contemp.

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An independent coalition in a graph G consists of two disjoint sets of vertices V 1 and V 2 neither of which is an independent dominating set but whose union V 1 ∪ V 2 is an independent dominating set. An independent coalition partition, abbreviated, ic-partition, in a graph G is a vertex partition π = {V 1 , V 2 , . . . , V k } such that each set V i of π either is a singleton dominating set, or is not an independent dominating set but forms an independent coalition with another set V j ∈ π. The maximum number of classes of an ic-partition of G is the independent coalition number of G, denoted by IC(G). In this paper we study the concept of ic-partition. In particular, we discuss the possibility of the existence of ic-partitions in graphs and introduce a family of graphs for which no ic-partition exists. We also determine the independent coalition number of some classes of graphs and investigate graphs G of order n with IC(G) ∈ {1, 2, 3, 4, n} and the trees T of order n with IC(T ) = n − 1.

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A note on the k-tuple domination number of graphs

Cited by 4 publications

References 4 publications

Some new results on the k-tuple domination number of graphs

Some new results on the k-tuple domination number of graphs

On Proper 2-Dominating Sets in Graphs

Independent coalition in graphs: existence and characterization

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